So, this is weird.  If I double click on the file from the windows folder, Access opens an gives me the above warning.  HOWEVER, if I open Access from the Start Menu, when Access opens and I double click on the recent files, the database opens without any issues.  So, I think that the problem is solved?

I have a new found respect for computer programmers.  This experience has been a good one...

Just to double check / rule out the obvious - go to each of the machines.    
        Open Access and click File -> Account -> About Access

That will bring up a dialog box showing your installation of Office.  

I would almost bet $$ that you have #4 installed as a different "bitness".

I'm assuming that when you created the ACCDE file you chose 64 bit AND you are using the exact same file to distribute to each computer.  Correct?

I just checked, and, yes sir, that was the issue. I never would have thought of that.  It is interesting, though, because I can open the database if I start Access first; I cannot open Access by double clicking the database.  Computers are interesting.

Darn - Should have put a $10 bet down.  Oh well  LOL

Glad that got it sorted.  If it were me I would completely uninstall Office on that computer then re-install 64 bit.  Kind of a hassle but  you will be rewarded for the effort on time saved in the future.

Word of the day: Bitness.

Ima open a new bitness and sell laptops.

Hey whatever works right?  Beau and You both knew what I was talking about so I guess I got the thought across.
LOL

Okay, so update:

To clarify, the .accdb file will open on any machine regardless of bitness.  The .accde file, however, will only open on a like-bitness machine.  It's kind of annoying, but my workaround on this is to save the .accde file on my machine & distribute it to the others; then, go to the 64-bit machine, open the "admin" .accdb file and save it as an .accde file directly to the 64-bit machine.  It's an extra step, but it's the best solution that I have thus far to make sure everyone can use the split database.

If anyone has a better solution, I am all ears (and, no, reenstalling 64-bit Office suite on all of the computers is not a solution for reasons that are above my paygrade).

Unfortunately, if you can't ensure that all of the machines are running the same version - 64 bit, the solution you just laid out is pretty much your only option.
Richard has a template for pushing updates - Access Updater v 1.0  but I don't think it accounts for mixes of bitness  ;)

Sorry, but that is definitely a Microsoft-imposed issue with ACCDE files. They require the same bitness. ACCDB files don't always.